We have seen that there is an adiabatic invariant associated with the
periodic gyration of a charged particle around magnetic field-lines.
Thus, it is reasonable to suppose that there is a second adiabatic invariant
associated with the periodic bouncing motion of a particle trapped
between two mirror points on
a magnetic field-line. This is indeed the case.

Recall that an adiabatic invariant is the lowest order approximation
to a Poincaré invariant:

(145)

In this case, let the curve correspond to the trajectory of
a guiding centre as a charged
particle trapped in the
Earth's magnetic field
executes a bounce orbit. Of course, this trajectory does not quite close,
because of the slow azimuthal drift of particles around the Earth. However,
it is easily demonstrated that
the azimuthal displacement of the end point of the trajectory, with respect to
the beginning point, is of order the gyroradius. Thus, in the limit
in which the ratio of the gyroradius, , to the variation length-scale of the
magnetic field, , tends to zero, the trajectory of the guiding centre
can be regarded as being approximately closed,
and the actual particle trajectory conforms very closely
to that of the guiding centre. Thus, the adiabatic invariant associated with
the bounce motion can be written

(146)

where the path of integration is along a field-line: from the equator to
the upper mirror point, back along the field-line to the lower mirror point, and
then back
to the equator. Furthermore, is an element of arc-length along the
field-line, and
.
Using
, the above
expression yields

(147)

Here, is the total magnetic flux enclosed by the curve--which,
in this case, is obviously zero. Thus, the so-called second adiabatic
invariant or longitudinal adiabatic invariant takes the form

(148)

In other words, the second invariant is proportional to the loop integral
of the parallel (to the magnetic field) velocity taken over a bounce orbit.
Actually, the above ``proof'' is not particularly rigorous: the rigorous proof
that is an adiabatic invariant was first given by Northrop and Teller. It should
be noted, of course, that is only a constant of the motion
for particles trapped in the inner magnetosphere provided that the
magnetospheric
magnetic field varies on time-scales much longer than the bounce time,
. Since the bounce time for MeV energy protons and electrons is,
at most, a few seconds, this is not a particularly onerous constraint.

Figure 4:The distortion of the Earth's magnetic field by the solar wind.

The invariance of is of great importance for charged particle
dynamics in the Earth's inner magnetosphere. It turns out that the
Earth's magnetic field is distorted from pure axisymmetry by the action of
the solar wind, as illustrated in Fig. 4. Because of this asymmetry, there
is no particular reason to believe that a particle will return to its
earlier trajectory as it makes a full rotation around the Earth. In other words,
the particle may well end up on a different field-line when it returns to
the same azimuthal angle. However, at a given azimuthal angle, each
field-line has a different length between mirror points, and a different
variation of the field-strength between the mirror points, for a particle
with given energy and magnetic moment . Thus, each field-line
represents a different value of for that particle. So, if is
conserved, as well as and , then the particle
must return to the same field-line after precessing around the Earth.
In other words, the conservation of prevents charged particles from
spiraling radially in or out of the Van Allen belts as they rotate around the Earth.
This helps to explain the persistence of these belts.